Lord Justice Leveson has said culture secretary Jeremy Hunt, his former special adviser Adam Smith and News Corp lobbyist Frédéric Michel would all give evidence to the inquiry into press ethics in the next fortnight.

The judge's announcement came as he called on parliament not to demand publication of written evidence submitted to him relating to the controversial emails written by Michel, which prompted calls for Hunt to resign.

Making a special address to his own inquiry this afternoon, Leveson asked that MPs wait for the three to give evidence "by the end of May". He asked that Commons Speaker John Bercow did not insist on the publication of fresh documentation before the cabinet minster appears.

Although Bercow can use a Speaker's order to force disclosure, Leveson said he hoped that there would be "sufficient respect" for the Leveson inquiry to prevent "disruption" of its work.

Leveson made his statement in response to a declaration on Monday by the Speaker who said parliament should be pre-eminent and that any "written documents" supplied by the minister to the inquiry should be shared with the Commons as "a courtesy to the house".

He warned MPs that if the Speaker or the Commons insisted on publishing evidence sent by the culture secretary before it he had appeared at the Leveson inquiry, he might be forced to abandon the questioning of Hunt, Michel and Smith on their roles in News Corp's £8bn bid for BSkyB.

The Labour party has already called for Hunt to resign, branding Michel's emails written to his boss James Murdoch as evidence that the culture secretary was biassed toward Rupert Murdoch in relation to the proposed takeover.

Hunt denied that he was party to any improper briefing of News Corp, but his special adviser Adam Smith resigned last month. Michel said most of his actual communication was with the special adviser, and Hunt concluded that the "volume and tone" of those contacts were inappropriate.

Leveson has already refused any attempt by No 10 to get Hunt's testimony brought forward and on Tuesday he said he will approach the inquiry from a "non-partisan judicial perspective". He reiterated his determination not to be drawn into a political debate about the emails released by James Murdoch as part of his testimony into the inquiry.

The judge said he was approaching the subject in a "neutral" fashion and although his investigations may look at the ministerial code and its efficacy, he would not be making any judgment on the issue. "I do not intend to consider, let alone adjudicate, on the issue of whether the house has been mislead," said Leveson.

Leveson says he it is appropriate to look at the BSkyB bid including the emails from Michel but said that "it is a matter for parliament to decide how far either the secretary of state or anyone else should go," in terms of disclosure.

But he said it was important that evidence was released to the inquiry first as this would mean no core witnesses would testify to the backdrop of a politically partisan debate. "The inquiry permits the public examination of this material in an independent impartial manner, visible to all as it happens," says Leveson.

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